Although John's answer is quite comprehensive, I would like to add this answer in order to reinforce my qualitative understanding of the matter and to try to provide the OP a more intuitive and qualitative explanation for the negative specific heat capacity as the OP seems to be looking for a more qualitative (and intuitive) sort of explanation.
For usual objects like rocks and stars, the temperature is a direct measure of the internal kinetic energy of the object - i.e., the kinetic energy of its constituents. Now, if - the configuration of such an object be of such a nature that whenever the internal kinetic energy increases (decreases), the structure of the object has to change in a way that makes its potential energy decrease (increase) by an amount greater than the increase (decrease) in its internal kinetic energy - then clearly the specific heat capacity will be negative!
For black holes, the story is a bit different. I haven't studied the work that determines Hawking temperature using the string theoretic microstates of a black hole and thus, I believe I can't really provide an explanatory or a deeper reasoning behind the negative specific heat capacity of black holes - but I will elucidate the mechanism of deriving the specific heat capacity of a black hole and that clearly shows that it must be negative.
The temperature of a black hole is given by $T = \dfrac{\hbar c^3}{8\pi GM}$. The energy of a black hole is to be considered as $E = Mc^2$. Therefore, $dE = -\dfrac{\hbar c^5}{8 \pi G T^2} dT$. Thus, specific heat capacity $C = \dfrac{1}{M}\dfrac{dE}{dT} = -\dfrac{\hbar c^5}{8 \pi GM T^2}$. In a qualitative way, one can also think that since the temperature of a black hole is bound to decrease with an increase in its area (larger the black hole, the cooler it is) and the area is bound to increase with an increase in its mass (energy), the specific heat capacity of the black hole has got to be negative.